Help with a seemingly trivial proof

2011-01-19T03:18:57Z

I have taken discrete logic/maths sophomore year of college (CS major), since then I have transferred and they are making me take something very similar (Foundations of Computer Science). Today my teacher was just going through some of the basics which I know pretty well. She then proceeded to write a proposition to prove that I am seeming to struggle with (strange I was good at these in the past). She was not clear on what method she wanted us to use or anything really, the question is my interpretation of what I think she wants us to prove, her accent or writing isn't to helpful :-. The question is as follows: *Prove x is odd if and only if 8 divides $x^2 - 1$, in more formally: $x\ is\ odd \iff 8|x^2-1$. Right now I am using the idea that d|n then n = dk.

So I basically started like this (Is contrapositive the way to go? modulus? -- she really didn't say much about these things it was the first day)

$8k = x^2 - 1$

$8k = 4n^2 - 1$

$(2n + 1)^2 - 1 = 8k$

$4n^2 + 4n = 8k$

$n^2 + n = 2k$ (divided by four)

$n(n + 1) = 2k$ (stuck)

I am clearly rusty on this stuff, if someone could at least point me in the correct direction it would be grateful, or provide me with a very similar example proof.

Thanks!

Help with a seemingly trivial proof - MathOverflow [closed]

Help with a seemingly trivial proof